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If P(x ,y ,z) is a point on the line ...

If `P(x ,y ,z)` is a point on the line segment joining `Q(2,2,4)a n d R(3,5,6)` such that the projections of ` vec O P` on te axes are 13/5, 19/5 and 26/5, respectively, then find the ratio in which `P` divides `Q Rdot`

Text Solution

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Since `vec(OP)` has projections `(13)/(5), (19)/(5) and (26)/(5)` on the coordinates axes, `vec(OP)=(13)/(5)hati+(19)/(5)hati+(26)/(5)hatk`. Suppose P divides the join of `Q(2, 2, 4) and R(3, 5, 6)` in the ratio `lamda:1`. Then the position vector of P is
`" "((3lamda+2)/(lamda+1))hati+((5lamda+2)/(lamda+1))hatj+((6lamda+4)/(lamda+1))hatk`
`therefore" "(13)/(5)hati+(19)/(5) hatj+(26)/(5) hatk`
`" "((3lamda+2)/(lamda+1))hati+((5lamda+2)/(lamda+1))hatj+((6lamda+4)/(lamda+1))hatk`
Thus, we have
`" "(3lamda+2)/(lamda+1)=(13)/(5), (5lamda+2)/(lamda+1)=(19)/(5) and (6lamda+4)/(lamda+1)=(26)/(5)`
`rArr" "2lamda=3 or lamda = 3//2`
Hence, P divides QR in the ratio 3 : 2.
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